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1b) Weak instruments are a third variable, Z, used when you have endogenous variables—variables that are influenced by other variables in the model. Instruments are used to account for unexpected behavior between variables. You are using them to find the true correlation between the explanatory variable (x) and response variable, (y). Therefore, you want these instruments to be as strong as possible. And also: Weak instruments—instruments that are only marginally valid—can cause many problems, including:
Biased estimates for independent variables,
Hypothesis tests with large size distortions
1c) A natural experiment is an empirical study in which the experimental conditions are determined by nature or by other factors out of the control of the experimenters and yet the treatment assignment process is arguably exogenous. Thus, natural experiments are observational studies and are not controlled in the traditional sense of a randomized experiment. Natural experiments are most useful when there has been a clearly defined and large change in the treatment to a clearly defined subpopulation, so that changes in responses may be plausibly attributed to the change in treatments.
As in natural experiments, the instrument is used to exploit an exogenous source of variation—created by explicit random assignment in these cases—to estimate the effect of interest. The use of such researcher-generated instruments is growing and reflects the
accelerating convergence of classical experimentation and observational research
methods.
Table 1 provides a sampling of some recent studies that have used instrumental variables techniques to analyze a natural experiment or a researcher generated randomized experiment. The first panel in Table 1 illustrates the breadth of application of the natural experiments idea in recent empirical work. Some of the examples are more convincing than others. But all are distinguished by a serious attempt to substantiate the underlying assumptions used to infer causality. There is more “theory” behind these attempts than in many ostensibly structural models where the justification for including or excluding certain variables is neither explicitly described nor evaluated.
1d)
The IV strategy is used to estimate causal relationship when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term, in which case OLS and ANOVA give biased results. A valid instrument induces changes in the explanatory variable but has no independent effect on the dependent variable, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable. The instrument must be correlated with the endogenous explanatory variables, conditionally on the other covariates. If this correlation is strong, then the instrument is said to have a strong first stage. A weak correlation may provide misleading inferences about parameter estimates and standard errors.
The instrument cannot be correlated with the error term in the explanatory equation, conditionally on the other covariates. In other words, the instrument cannot suffer from the same problem as the original predicting variable. If this condition is met, then the instrument is said to satisfy the exclusion restriction.
2a) Mothers that smoke during pregnancy might as well be less careful with other health issues that affect birth weight and that are not controlled for in the regression equation
2b) If people choose the state of residence independently from the price of cigarettes, which seems a sensible assumption, then the price of cigarettes might be uncorrelated with birth weight through ways other than the amount of cigarettes smoked. In this case, the IV identification assumption, that the instrument and the error term are uncorrelated, will hold. However, we may as well suspect that smokers are more sensitive to health problems that may require treatment and may affect birth weight. If this is true, and price affects consumption, then the exclusion restriction ceases to hold. As for the rank condition, it depends on whether the instrument (price) has enough variation across states to actually affect consumption levels.
2c) bcuse bwght.dta
sum
gen lbw=ln( bw)
gen ly=ln( y)
regress lbw cig male order ly
ivreg lbw (cig=cigprice) male order ly
The OLS results show that the amount of cigarettes smoked while pregnant significantly reduce birth weight by 8%. However, the IV estimates show a non-significant impact of the amount of cigarettes on birth weight. If the IV estimates are correct, all impact seems to come from how smoking is related to other health issues or behaviour that affects birth weight, not through smoking directly. However, the IV estimates show a worrying feature: all coefficients become insignificant, not only the one on the amount of cigarettes smoked while pregnant. We know that IV estimates have generally a higher variance than the OLS ones, but it may become a problem if the instruments are only weakly correlated with the endogenous explanatory variables. If this is so, the IV estimator becomes inconsistent and its variance becomes very large.
2d)
regress cig cigprice male order ly
What we predicted in the last question turns out to be true. The reduced for model for cig shows that the instrument cigprice is weak, not explaining the endogenous variables.
3a)
Suppose we use the reduced form for w
w = z2γ + ν
and estimate γ through OLS. Then E (z2 ν (^γ)) = 0 by construction but it is possible that
E (z1 ν (^γ)) is not equal to 0.
If the inequality holds, then the estimates will be inconsistent. To see why, notice that
y = z1β + wα + e
= z1β + (z2^γ + ^ν) α + e
= z1β + ^wα + e + ^να
= z1β + ^wα + v
where ^ν = w − ^w and ^w = z2γb. The problem is that E (z’1 ν) is not equal to 0 and it leads to inconsistency.
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